Triangle Similarity Theorem Identification: Finding the ABC-DEF Ratio

To determine which theorem proves the triangles are similar, we'll use the Angle-Angle (AA) Similarity Theorem:

• Step 1: Check the angles $\angle ABC$ and $\angle DEF$, and $\angle ACB$ and $\angle DFE$.
• Step 2: Since the problem implies these angles are equal, the AA criterion confirms the triangles are similar.

Next, we calculate the ratio of similarity:

• Step 3: Identify the corresponding sides, such as $AB$ and $ED$, $BC$ and $DF$, and $AC$ and $EF$.
• Step 4: Establish the correct ratio:
  $\frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF}$

Therefore, according to the AA similarity theorem, the triangles are similar with the ratio of similarity $\frac{AB}{ED}=\frac{BC}{DF}=\frac{AC}{EF}$.

The correct choice is:

AA, $\frac{AB}{ED}=\frac{BC}{DF}=\frac{AC}{EF}$